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The million Dollar questions

If you'd like to have a go at solving one of the most tricky and important open questions in maths and get rich in the process, then the QEDen web site is worth a look. The site has been set up with the ambitious goal to solve the Millennium Prize Problems, seven of the most
difficult unsolved problems in maths. The Clay Mathematics Institute has promised a million Dollars to anyone who solves one of the problems, but so far solutions remain elusive.

QEDen is designed to be "an online playground for the mathematically and scientifically minded people on the internet to converge and wrap their minds around the toughest problems yet to face the planet". As yet, its "progress thermometer" is firmly stuck at zero. But, who knows, maybe you could deliver a vital clue. And, don't worry, if you solve the problem, all the money is yours
alone.

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Maths maps musical emotion

Music makes emotion and maths underpins music. But can the emotional part of music be captured by maths? To find the answer to this question Elaine Chew, assistant professor of industrial and systems engineering at the USC Viterbi School of Engineering, set up a graduate course called "Computational Modelling of Expressive Performance" to look for answers. The aim of the course is to teach
students how to use tools from engineering and computation to understand our emotional responses to music.

Chew pursues a successful career as a pianist along side her research, and so is perfectly qualified. But as she admits, teaching the course is a challenge, as students need expertise from a huge range of different disciplines. They need to use tools from music theory, cognitive science, artificial intelligence, experimental psychology, signal processing and neuroscience, to name but a few.
Not many students have the necessary pre-requisites, but some of those that do have now put an exhibition of their projects online.

One pair of students have created "emotiongrams" that map musical characteristics to colour patterns to capture their emotional impact. Another project involves analysing scores of silent movies that have been specifically created to evoke certain emotions, and yet another one involves a system that allows you to add extra rules to a musical score and observe whether this changes the impact on
your feelings.

You can read about all the projects on the course website. Plus has a number of articles on the mathematics of music, which you can find in our archive under keyword or topic "maths and music".

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29 steps to universal (26/04/2006)

Think of a number, any number, and see if you can write it as the sum of square numbers: 13 = 22 + 32, 271 = 12 + 12 + 102 + 132, 4897582 = 62 + 952 + 22112...

In 1770, Lagrange proved that every positive integer, no matter how large, can be written as a sum of at most four squares, x2 + y2 + z2 + t2. In the centuries since, mathematicians searched for other universal quadratic forms which could represent
all the positive integers. Another 53 expressions, including 1x2 + 2y2 + 3z2 + 5t2, were found by Ramanujan in 1916. So many such universal forms exist, but how can you predict if a particular quadratic form is universal?

Now the brilliant young mathematician Manjul Bhargava and his colleague Jonathan Hanke have found a surprisingly simple result that completely solves the problem of finding and understanding universal quadratic forms. They found a shortcut to deciding if a quadratic form is universal — to check if the form represents every single positive integer, you only have to check it represents a mere 29
particular integers, the largest of which is 290. Bharghava and Hanke then went on to find every universal quadratic form (with four variables), all 6,436 of them. You can read about this surprising result in Ivars Peterson's excellent article in Science News.

Apart from Bhargava's brilliance as a mathematician (he was one of the youngest people to be made a full professor at just 28), he is also an accomplished musician. Both number theory and tabla playing may be viewed as the study of patterns, Bhargava told Peterson. "The goal of every number theorist and every tabla player," he explains in the article, "is to combine these patterns, carefully
and creatively, so that they flow as a sequence of ideas, tell a story, and form a complete and beautiful piece."

If you're struggling with a sum of squares, try out this useful applet by Dario Alpern.

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Frequently flipping fields (29/3/2006)

In the past news story, Core business, we reported on research that the Earth's magnetic field may be in the early stages of a polarity reversal. These geomagnetic flips have occurred hundreds of times in the last 160 million years, and as the last one was 780,000 years ago many researchers think we are overdue for the next reversal. Until
recently it was thought that the frequency of these polarity reversals was random, following a Poisson distribution. However Italian physicists have found that the process is better described by a Levy distribution — a distribution used for processes that exhibit "memory" of past events. So the pole reversals are not independent events,
and this information should inform models of the magnetic dynamo process believed responsible. But as for when the next flip is due, that still remains a mystery.

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Happy Pi Day to you! (14/3/2006)

Write the date March 14th in the date format used in the US and you get 3.14 — which makes it Pi Day! Plus is celebrating by reflecting on the ubiquitous usefulness of this number to mathematics, pondering its many unsolved mysteries, and
of course eating an appropriately shaped pi pie (pie are squared you know). How will you be celebrating? If reciting digits of pi is to be your pi party trick, then make sure you read Remembrance of numbers past from issue 31.

"If a billion decimals of pi were printed in ordinary type, they would stretch from New York City to the middle of Kansas. Only fortyseven decimal places of pi would be sufficiently precise to inscribe a circle around the visible universe that didn't deviate from perfect circularity by more than the distance across a
single proton."
- from The Riciculously Enhanced Pi Page

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Gravity kills dark energy? (16/02/2006)

In 1998, scientists found that the rate at which the universe expands is accelerating. This was extremely puzzling, as none of the standard theories could account for the acceleration. So scientists decided that something unknown and mysterious must be responsible, and they called it dark energy. But now three scientists claim that we may not need dark energy after all, but that the
acceleration can be explained by a modified model of gravity. They altered the equations from the theory of general relativity in a way prescribed by cosmological theories based on a universe with more than four dimensions. When applied to short distances (on the cosmological scale), the differences between the old and new models are not noticeable, however, on ultra large distances the new
equations predict the accelerated expansion. Moreover, the researchers say, the predictions match observations extremely well and don't conflict with other observations.